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Let X={X_n:n=0,1,2,...} be an irreducible, positive recurrent Markov chainwith invariant probability measure \pi. We show that if X satisfies a one-stepminorization condition, then \pi can be represented as an infinite mixture. Thedistributions in the mixture are associated with the hitting times on anaccessible atom introduced via the splitting construction of Athreya and Ney[Trans. Amer. Math. Soc. 245 (1978) 493-501] and Nummelin [Z. Wahrsch. Verw.Gebiete 43 (1978) 309-318]. When the small set in the minorization condition isthe entire state space, our mixture representation of \pi reduces to a simpleformula, first derived by Breyer and Roberts [Methodol. Comput. Appl. Probab. 3(2001) 161-177] from which samples can be easily drawn. Despite the fact thatthe derivation of this formula involves no coupling or backward simulationarguments, the formula can be used to reconstruct perfect sampling algorithmsbased on coupling from the pas

A mixture representation of π with applications in Markov chain Monte Carlo and perfect sampling